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Properties

Label 2-2303-2303.1362-c0-0-4
Degree $2$
Conductor $2303$
Sign $0.00641 + 0.999i$
Analytic cond. $1.14934$
Root an. cond. $1.07207$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.206 + 0.0311i)2-s + (1.41 − 0.967i)3-s + (−0.913 − 0.281i)4-s + (0.323 − 0.155i)6-s + (0.712 − 0.701i)7-s + (−0.368 − 0.177i)8-s + (0.711 − 1.81i)9-s + (−1.56 + 0.483i)12-s + (0.169 − 0.122i)14-s + (0.719 + 0.490i)16-s + (0.411 + 0.381i)17-s + (0.203 − 0.352i)18-s + (0.331 − 1.68i)21-s + (−0.694 + 0.104i)24-s + (−0.988 + 0.149i)25-s + ⋯
L(s)  = 1  + (0.206 + 0.0311i)2-s + (1.41 − 0.967i)3-s + (−0.913 − 0.281i)4-s + (0.323 − 0.155i)6-s + (0.712 − 0.701i)7-s + (−0.368 − 0.177i)8-s + (0.711 − 1.81i)9-s + (−1.56 + 0.483i)12-s + (0.169 − 0.122i)14-s + (0.719 + 0.490i)16-s + (0.411 + 0.381i)17-s + (0.203 − 0.352i)18-s + (0.331 − 1.68i)21-s + (−0.694 + 0.104i)24-s + (−0.988 + 0.149i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00641 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2303\)    =    \(7^{2} \cdot 47\)
Sign: $0.00641 + 0.999i$
Analytic conductor: \(1.14934\)
Root analytic conductor: \(1.07207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2303} (1362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2303,\ (\ :0),\ 0.00641 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.821752982\)
\(L(\frac12)\) \(\approx\) \(1.821752982\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.712 + 0.701i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
good2 \( 1 + (-0.206 - 0.0311i)T + (0.955 + 0.294i)T^{2} \)
3 \( 1 + (-1.41 + 0.967i)T + (0.365 - 0.930i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.733 - 0.680i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.411 - 0.381i)T + (0.0747 + 0.997i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.313 - 0.0965i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (-0.369 - 0.113i)T + (0.826 + 0.563i)T^{2} \)
59 \( 1 + (-0.0785 + 1.04i)T + (-0.988 - 0.149i)T^{2} \)
61 \( 1 + (1.61 - 0.496i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.388 - 1.70i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.134 + 0.232i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.591 - 1.50i)T + (-0.733 - 0.680i)T^{2} \)
97 \( 1 - 1.82T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.784960177487861677405864533146, −8.191859974494205013815819306922, −7.70520400640278027396917723761, −6.89524660438944223197239835506, −5.92558214551116713654557776191, −4.87842833830845974492506604514, −3.94079217194380928346073156873, −3.33399348385974737637822938336, −2.02925314618597204469035902683, −1.13249271994782776773910491577, 1.94179155411931487638479142459, 2.97674860773677895238447585557, 3.63892112004551342348541125505, 4.54910672890111819080996248363, 5.02696165767891240871559379566, 6.00642284637948622659018561160, 7.67832907521137812742057053950, 7.932056854526066502790020121585, 8.882593352343370959854280572343, 9.098119620627149077460884837201

Graph of the $Z$-function along the critical line